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In this lesson, students will gain a better understanding of the parameters of the general form of a sine equation [**y = a sin(k(x – d)) + c**], where **a** is amplitude (vertical stretch or compression), **k** is period (horizontal stretch or compression), **d** is phase shift (horizontal displacement), and **c** is vertical displacement. Students will retrieve data from Statistics Canada's E-STAT database for one of Canada's youth cohorts (ages 20 to 24) and import these data into a statistical software program. Within the software program, students will model a sine function. By adjusting the values of the **a**, **k**, **d**, and **c** parameters to maximize the visual fit of the curve to the data, students will learn about the purpose of each variable in the equation.

**Contributors:** Jennifer Hall and Joel Yan, Statistics Canada; Heather Curl, Sarnia Collegiate Institute and Technical School; Jennifer Brown, St. Michael Catholic High School, Kemptville

- Describe key properties of periodic functions arising from real-world applications, given a numeric or graphical representation
- Determine the amplitude, period, phase shift, displacement, and range of sine functions whose equations are given in the form
**y = a sin(k(x – d)) + c** - Identify restrictions that the context of data places on domain and range
- Represent a sine function with an equation, given its graph
- Through investigation, determine how sine functions can be used to model periodic phenomena that do not involve angles

Grades 11 and 12

Mathematics

One to two 75 minute periods

- Computers with Internet access and statistical software
- E-STAT account
- Computer projector
- E-STAT instructions
- Software instructions:
- Student worksheet
- Student worksheet – Teacher version

Sine equation in **y = a sin(k(x – d)) + c** form

Basic knowledge of E-STAT and statistical software

- Discuss important properties of the general form of sinusoidal equations as a review.
- Using the computer projector, demonstrate the important features of E-STAT. You may wish to use the flash presentation What's E-STAT?
- Hold a brief class discussion on the topic of the Baby Boom to assess students' prior knowledge and share information on the topic.
- Distribute the student instructions and worksheet and have students complete the lesson independently or in pairs.

Have students fit a cosine curve to the data instead of a sine curve. Prior to plotting the curve, have students hypothesize what variable(s) will change. Have students summarize the differences between sine and cosine curves.

Have students perform trigonometric regression analysis on these data in either a software program or graphing calculator. Have them examine the r^{2} value to see how well the data fit a sine curve. Have the students compare their curve of best fit to the curve from the regression analysis.

Have students repeat the lesson using the 15 to 19 year-old age cohort instead. Have students note differences in the shape of the sine curve and attempt to explain these differences using sociological reasons.

Have students retrieve data on the number of births in the post-World War II Baby Boom period. Have them compare the shape of this graph to their 20 to 24 year-old youth cohort graph and note similarities and differences.

Students can be informally assessed on their work habits and computer skills throughout this activity. They can be formally assessed via the worksheet, which can be handed in to be marked using a marking scheme of the teacher's choice.